Fairness and Optimal Stochastic Control for Heterogeneous Networks

Transcription

1 λ 91 λ 93 Fairness and Optimal Stochastic Control for Heterogeneous Networks sensor network wired network wireless λ 48 λ λ n R n U n Michael J. Neely (USC) Eytan Modiano (MIT) Chih-Ping Li (USC) Time-Varying Channels

2 A heterogeneous network with N nodes and L links: λ 91 λ 93 sensor network wired network wireless λ 48 λ Γ S = channel dependent set of transmission rate matrices Γ S = Γ Α SA Γ Β Γ C SC λ n R n U n Choose µ Γ S Slotted time t = 0, 1, 2, t Traffic (A ij ) and channel states S i.i.d. over timeslots

3 A heterogeneous network with N nodes and L links: λ 91 λ 93 sensor network wired network wireless λ 48 λ Γ S = channel dependent set of transmission rate vectors Γ S = Γ Α SA Γ Β Γ C SC λ n R n U n Choose µ Γ S Input rate matrix: (λ ij ) (where E[A ij ] = λ ij ) Channel state vector: S = (S 1, S 2,, S L ) Transmission rate vector: µ = (µ 1, µ 2,, µ L ) Resource allocation: choose µ Γ S

4 Goal: Develop joint flow control, routing, resource allocation sensor network wired network wireless λ λ λ 1 λ n λ 48 λ 42 R n U n λ 2 Λ = Capacity region (considering all routing, resource alloc. policies) g nc (r nc ) = concave utility functions util r

5 Some precedents: Static optimization: (Lagrange multipliers and convex duality) Kelly, Maulloo, Tan, Oper Res [pricing for net. optimization] Xiao, Johansson, Boyd, Allerton 2001 [network resource opt.] Julian, Chian, O Neill, Boyd, Infocom 2002 [static wireless opt] Lee, Mazumdar, Shroff, Infocom 2002 [static wireless downlink] Marbach, Infocom 2002 [pricing, fairness static nets] Krishnamachari, Ordonez, VTC 2003 [static sensor nets] Low, TON 2003 [internet congestion control] Dynamic control: D. Tse, 97, 99 [ proportional fair algorithm: max U i /r i ] Kushner, Whiting, Allerton 2002 [ prop. fair alg. analysis] S. Borst, Infocom 2003 [downlink fairness for infinite # users] Li, Goldsmith, IT 2001 [broadcast downlink] Tsibonis, Georgiadis, Tassiulas, Infocom 2003 [max thruput outside of capacity region]

6 Stochastic Stability via Lyapunov Drift: Tassiulas, Ephremides, AC 1992, IT 1993 [MWM, Diff. backlog] Andrews et. al., Comm. Mag, 2003 [server selection] Neely, Modiano, Rohrs, TON 2003, JSAC 2005 [satellite, wireless] McKeown, Anantharam, Walrand, Infocom 1996 [NxN switch] Leonardi et. Al., Infocom 2001 [NxN switch]

7 Example: Server alloc., 2 queue downlink, ON/OFF channels λ 2 Pr[ON] = p λ 1 λ 2 Pr[ON] = p 2 λ 1 Capacity region Λ: 0.5 MWM algorithm (choose ON queue with largest backlog) Stabilizes whenever rates are strictly interior to Λ [Tassiulas, Ephremides IT 1993]

8 Comparison of previous algorithms: (1) MWM (max U i µ i ) (2) Borst Alg. [Borst Infocom 2003] (max µ i /µ i ) (3) Tse Alg. [Tse 97, 99, Kush 2002] (max µ i /r i )

9 sensor network wired network wireless λ λ λ n λ 48 λ 42 R n U n Approach: Put all data in a reservoir before sending into network. Reservoir valve determines R n (amount delivered to network from reservoir (n,c) at slot t). Optimize dynamic decisions over all possible valve control policies, network resource allocations, routing to provide optimal fairness.

10 Part 1: Optimization with infinite demand sensor network wired network wireless λ λ λ 1 λ n λ 48 λ 42 R n U n λ 2 Assume all active sessions infinitely backlogged (general case of arbitrary traffic treated in part 2).

11 Cross Layer Control Algorithm (CLC1): (1) Flow Control: At node n, observe queue backlogs U n for all active sessions c. λ (c1) n R (c1) n Rest of Network R (c2) n λ n (c2) U n (where V is a parameter that affects network delay)

12 (2) Routing and Scheduling: link l c l * = ( (similar to the original Tassiulas differential backlog routing policy [1992]) (3) Resource Allocation: Observe channel states S. Allocate resources to yield rates µ such that: Maximize: l W l* µ l Such that: µ Γ S

13 Theorem: If channel states are i.i.d., then for any V>0 and any rate vector λ (inside or outside of Λ), λ 1 µ sym optimal point r * µ sym Avg. delay: Fairness: (where )

14 Special cases: (for simplicity, assume only 1 active session per node) 1. Maximum throughput and the threshold rule Linear utilities: g nc (r) = α nc r λ n R n U n (threshold structure similar to Tsibonis [Infocom 2003] for a downlink with service envelopes)

15 (2) Proportional Fairness and the 1/U rule logarithmic utilities: g nc (r) = log(1 + r nc ) λ n R n U n

16 Mechanism Design and Network Pricing: greedy users each naturally solves the following: Maximize: g nc (r) - PRICE nc r Such that : 0 r R max This is exactly the same algorithm if we use the following dynamic pricing strategy: PRICE nc = U nc /V

17 Analytical technique: Lyapunov Drift Lyapunov function: L(U) = n U n2 Lyapunov drift: Δ = E[L(U(t+1) - L(U) U] Theorem: (Lyapunov drift with Utility Maximization) If for all t: Δ C - ε n U n - VE[g(r ) U] - Vg(r *) Then: (a) n E[U n ] C + VNG max ε (stability and bounded delay) (b) g(r achieve ) + C/V (resulting utility) g(r *)

18 Part 2: Scheduling with arbitrary input rates λ 1 λ 2 λ n R n U n Novel technique of creating flow state variables Z nc Y nc = R max - R nc Z nc = max[z nc - g nc, 0] + Y nc (Reservoir buffer size arbitrary, possibly zero)

19 Cross Layer Control Alg. 2 (CLC2) the Z nc (t+1) iteration of the previous slide.

20 Simulation Results for CLC2: (i) 2 queue downlink a)g 1 (r)=g 2 (r)= log(1+r) λ 1 λ 2 Pr[ON] = p 1 Pr[ON] = p 2 b)g 1 (r)=log(1+r) g 2 (r)=1.28log(1+r) (priority service)

21 (ii) 3 x 3 packet switch under the crossbar constraint: proportionally fair

22 Concluding Slide: (iii) Multi-hop Heterogeneous Network λ 91 λ 93 sensor network wired network wireless λ 48 λ λ n R n U n λ 91 = λ 93 = λ 48 = λ 42 = 0.7 packets/slot (not supportable) The optimally fair point of this example can be solved in closed form: r 91 * = r 93 * = r 48 * = 1/6 = , r 42 = 0.5 Use CLC2, V= > U tot =858.9 packets r 91 = , r 93 =0.1662, r 48 =0.1678, r 42 =0.5000

23 The end

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